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Journal of Function Spaces

Volume 2014 (2014), Article ID 416187, 8 pages

http://dx.doi.org/10.1155/2014/416187
Research Article

Fixed Point Theorems for Contractions of Rational Type with PPF Dependence in Banach Spaces

1Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

2Department of Mathematics, Rzeszów University of Technology, Aleja Powstanców Warszawy 8, 35-959 Rzeszów, Poland

Received 10 April 2014; Accepted 24 May 2014; Published 9 June 2014

Academic Editor: Józef Banaś

Copyright © 2014 J. Rocha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the existence of the PPF dependent fixed point in the Razumikhin class for contractions of rational type in Banach spaces, by using a general class of pairs of functions. Our result has as particular cases a great number of interesting consequences which extend and generalize some results appearing in the literature.

1. Introduction

Banach’s contraction principle is one of the pivotal results of analysis. Its significance lies in its vast applicability to a great number of branches of mathematics and other sciences, for example, theory of existence of solutions for nonlinear differential, integral, and functional equations, variational inequalities, and optimization and approximation theory.

Generalizations of the contractive mapping theorem have been a heavily investigated branch of research. In particular, this principle was extended in [1], where the domain of the nonlinear operator involved is and the range is , where is a Banach space. This result is known as the contraction theorem for operator with PPF (past, present, and future) dependence. The PPF fixed point theorems are useful for proving the existence of solutions for nonlinear functional-differential and integral equations which may depend upon the past history, present data, and future considerations. Some papers about fixed point theorems with PPF dependence have appeared in the literature (see, e.g., [15]).

On the other hand, Dass and Gupta in [6] and Jaggi in [7] were the pioneers in proving fixed point theorems using contractive conditions involving rational expressions. In [4], the authors present a fixed point theorem for contractions of rational type with PPF dependence.

The purpose of this paper is to present a fixed point theorem for generalized contractions of rational type with PPF dependence which has, as particular cases, interesting consequences. Particularly, our result extends the one appearing in [4].

2. Preliminaries

Throughout this paper, will denote a Banach space with norm and will denote the space of the continuous -valued functions defined on and equipped with the norm given by

Let be a mapping. A point is said to be a PPF dependence fixed point of   or a fixed point with PPF dependence of   if   , for some .

For a fixed , the Razumikhin class is defined by

Remark 1. Notice that, for fixed, the function , defined by satisfies , for any and, therefore, for any . Consequently, for any .

We say that the class is algebraically closed with respect to difference if for any   we have   . Similarly, we say that the class is topologically closed if it is closed with respect to the topology on induced by the norm .

The Razumikhin class plays an important role in the existence of PPF fixed point.

The first result about the existence of PPF fixed point appears in [1] and it is presented in the following theorem.

Theorem 2 (see [1]). Suppose that is a mapping such that there exists satisfying for any   . If   is topologically closed and algebraically closed with respect to difference for some , then has a unique PPF dependent fixed point in .

Recently, in [4] the authors proved the following PPF dependent fixed point theorem for rational type contraction mappings.

Theorem 3 (see [4]). Let be a mapping satisfying for any and where with and . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

The main purpose of this paper is, by using a class of pairs of functions satisfying certain assumptions, to present new PPF dependent fixed point theorems for contractions of rational type. Particularly, our result generalizes the main result of  [4] (Theorem 3).

3. Main Results

We start this section presenting the following class of pairs of functions . A pair of functions is said to belong to the class if they satisfy the following conditions:(i) ;(ii)for , if , then ;(iii)for and sequences in such that if for any , then .

Remark 4. Notice that if and , then , since we can take for any and by (iii) we deduce .

In the sequel, we present some interesting examples of pairs of functions belonging to the class which will be very important in our study.

Example 5. Let be a continuous and increasing function such that if and only if (these functions are known in the literature as altering distance functions).

Let be a nondecreasing function such that if and only if   and suppose that .

Then the pair .

In fact, it is clear that satisfy (i).

To prove (ii), suppose that and . Then, from and taking into account the increasing character of , we can deduce that .

In order to prove (iii), we suppose that where and Taking in (8), we infer that .

Let us suppose that . Since , we can find and a subsequence of such that for any . As is nondecreasing, we have for any and, consequently, . This contradicts the fact that . Therefore, .

This proves that .

An interesting particular case is when is the identity mapping, , and is a nondecreasing function such that if and only if   and for any .

Example 6. Let be the class of functions defined by Let us consider the pairs of functions , where and is defined by , for .

Then .

It is clear that the pairs with satisfy (i).

To prove (ii), from we infer, since , that and, consequently, satisfies (ii).

In order to prove (iii), we suppose that where and .

Let us suppose that .

Since , we can find a subsequence such that for any . Now, as in particular, we have and, since for any , Taking in the last inequality, we obtain Finally, since , we infer that and this contradicts the fact that .

Therefore, .

This proves that for .

Remark 7. Suppose that is an increasing function and . Then it is easily seen that the pair .

Now, we are ready to present our main result.

Theorem 8. Let be a mapping such that there exists a pair of functions and such that for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Proof. Let be an arbitrary function in (whose existence is guaranteed by Remark 1). Since , put .

Again, by Remark 1, we can find such that Since , put . Using the same argument, we can find such that Repeating this process, we can obtain a sequence in such that Since is algebraically closed with respect to difference, we have for any .

First, we will prove that . In fact, taking into account (21) and (22), we get and, therefore, applying the contractive condition, we have

Let us suppose that there exists such that .

In this case, and, consequently, .

By (21), we have and would be the PPF dependent fixed point.

In the sequel we suppose that for any .

Now, we can distinguish two cases.

Case  1. Consider In this case, from (24), we infer and, since , we deduce Case  2. Consider In this case, from (24) and since , we infer By (21) and (22), we have Since , from the last inequality, it follows that and, therefore, In both cases, we obtain that inequality (28) is satisfied and, consequently, the sequence is a decreasing sequence of nonnegative real numbers.

Put , where , and denote We remark the following.(1)If   , then from (24) we can find infinitely many natural numbers satisfying inequality (27) and, since and , we deduce that .(2)If   , then from (24) we can find infinitely many such that Since and using a similar argument to the one used in Case 2, we obtain for infinitely many . Taking in the last inequality and taking into account that we deduce and, consequently, . Since , we obtain .

Therefore,

Next, we will prove that is a Cauchy sequence in . In contrary case, since , by Lemma 2.1 of  [8], we can find and subsequences and of satisfying Since for any , from (21) and (22), we have for any .

Using the contractive condition and (21) and (22), we obtain for any .

Let us put By (41) we have   or   .

Let us suppose that   . Then there exist infinitely many such that and since and we infer from (39) that . This is a contradiction.

Let us suppose that   . In this case, we can find infinitely many such that and since , we infer Taking   and in view of (38) and (39), it follows that and this is a contradiction.

Therefore, since in both possibilities     and   we obtain a contradiction, we deduce that is a Cauchy sequence in .

Since is a Banach space, we can find such that . As and is topologically closed, we have .

Next, we will prove that is a PPF dependent fixed point of . In fact, by the contractive condition, we obtain for any .

We can distinguish two cases again.(1)There exist infinitely many such that In this case, since , we obtain for infinitely many . Since , taking in the last inequality, we obtain where, to simplify our considerations, we will denote the subsequence by the same symbol . By (21), in ; this means that and, consequently, . From this last result and from (51) we deduce that and, therefore, is a PPF dependent fixed point of in .(2) There exist infinitely many such that To simplify our considerations, we will denote the subsequence by the same symbol . Since , we infer for any . Using (21), we have that and, therefore, for any . Taking and by (38) since , we infer (50). From the above case, we deduce that is a PPF dependent fixed point of in . Therefore, we have proved that in both cases is a PPF dependent fixed point of in .

Finally, we will prove the uniqueness of PPF dependent fixed point of in .

Suppose that is another PPF dependent fixed point of in . Then, since , and is algebraically closed, we obtain As and because and are PPF dependent fixed points of , we infer Consequently, using the contractive condition, we get We can distinguish two cases.(i)Consider . In this case, from (59) we have Now, since and using Remark 4, we get and, therefore, .(ii)Consider . From (59) we obtain and, since , we infer that . Therefore, or, equivalently, .

This result finishes the proof.

By Theorem 8, we obtain the following corollaries.

Corollary 9. Let be a mapping such that there exists a pair of functions and satisfying for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Corollary 10. Let be a mapping such that there exists a pair of functions and satisfying for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

The main result of [4] is Theorem 3. Notice that the contractive condition appearing in this theorem for any , where with and , implies that for any . This condition is a particular case of the contractive condition appearing in Theorem 8 with the pair of functions given by and . Therefore, Theorem 3 is a particular case of the following corollary.

Corollary 11. Let be a mapping such that there exist real numbers with and satisfying for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Taking into account Example 5, we have the following corollary.

Corollary 12. Let be a mapping such that there exist two functions and such that for any , where is a continuous and increasing function satisfying if and only if , and is a nondecreasing function such that if and only if , and .

If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Corollary 12 has the following consequences.

Corollary 13. Let be a mapping such that there exist two functions and such that for any , where is an increasing function and is a nondecreasing function and they satisfy if and only if , and is continuous with . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Corollary 13 can be considered as the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems which appears in [9].

Theorem 14 (see [9]). Let be a complete metric space and a mapping satisfying for , where and satisfy the same conditions as in Corollary 13. Then has a unique fixed point.

Corollary 15. Let be a mapping such that there exist two functions satisfying the same conditions as in Corollary 13 and such that for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Taking into account Example 6, we have the following corollary.

Corollary 16. Let be a mapping such that there exist (see Example 6) and satisfying for any . If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

A consequence of Corollary 16 is the following result.

Corollary 17. Let be a mapping such that there exists satisfying for any .

If such that is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

Corollary 17 is the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems appearing in [10].

Theorem 18. Let be a complete metric space and a mapping satisfying for any , where . Then has a unique fixed point.

Conflict of Interests

The authors declare that there is no conflict of interests in the submitted paper.

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